A={{1,2},{1,-1}};B=Eigenvectors[A];

DynamicModule[{v={1,0},axes=True,circles=False,stdcross=False,eigencross=False,arrow=True},
{Graphics[{Dynamic[If[arrow,Locator[Dynamic[v]],{}]],Dynamic[If[stdcross,{Line[{{-10,0},{10,0}}],Line[{{0,-10},{0,10}}]},{}]],Dynamic[If[eigencross,{Line[{-10 B[[1]],10 B[[1]]}],Line[{-10B[[2]],10B[[2]]}]},{}]],Dynamic[If[circles,Black,White]],Table[Line[Table[({5i,5j}+2{Sin[k],Cos[k]}),{k,0,2Pi+.2,.2}]],{i,-6,6},{j,-6,6}],Blue,Thick,Dynamic[If[arrow,Arrow[{{0,0},Dynamic[v]}],{}]]},Axes->Dynamic[axes],PlotRange->{{-10,10},{-10,10}},ImageSize->250],Graphics[{Dynamic[If[stdcross,{Line[{A.{-10,0},A.{10,0}}],Line[{A.{0,-10},A.{0,10}}]},{}]],
Dynamic[If[eigencross,{Line[{-10 B[[1]],10 B[[1]]}],Line[{-10B[[2]],10B[[2]]}]},{}]],Dynamic[If[circles,Black,White]],Table[Line[Table[{{1,2},{1,-1}}.({5i,5j}+2{Sin[k],Cos[k]}),{k,0,2Pi+.2,.2}]],{i,-6,6},{j,-6,6}],Blue,Thick,Dynamic[If[arrow,Arrow[{{0,0},A.v}],{}]]},Axes->Dynamic[axes],PlotRange->{{-10,10},{-10,10}},ImageSize->250],Panel[Column[{Row[{Checkbox[Dynamic[arrow]],"arrow"}],Row[{Checkbox[Dynamic[circles]],"background"}],Row[{Checkbox[Dynamic[axes]],"axes"}],Row[{Checkbox[Dynamic[stdcross]], "standard cross"}],Row[{Checkbox[Dynamic[eigencross]],"eigencross"}]}]]}]